|       MEIRlCAli 
ETERMINATIONDF  HEIGHTS 


THE  BAROMETRICAL 
DETERMINATION  OF  HEIGHTS 

A    PRACTICAL    METHOD 

OF 

BAROMETRICAL    LEVELLING 
AND    HYPSOMETRY 

FOR 

SURVEYORS  AND  MOUNTAIN  CLIMBERS 

BY 

F.    J.    B.    CORDEIRO 
t< 

Second  Edition,   Revised  and  Enlarged 

NEW  YORK 
SPOX  &  CHAMBERLAIN,  120  LIBERTY  ST. 

LONDON 

E.  &  F.  N.  SPOX,  LIMITED,  57  HAYMARKET 
1917 


Copyright,  1897 

Copyright,  1917,  by 

SPON  &  CHAMBERLAIN 


BARR  &  HAYFIELD,   Inc.,  Printers,  159  William  St.,  New  York 


PREFACE  TO  SECOND  EDITION. 


The  methods  and  formulas  given  in  this  book 
have  been  recognized  by  competent  authority  as 
the  only  ones  that  can  give  correct  and  consist- 
ent results.  The  older  formulas  are  incorrect, 
and  the  results  obtained  by  them  very  inconsist- 
ent. Although  this  book  has  been  in  print  some 
years,  attempts  are  still  made  to  determine  moun- 
tain heights  by  antiquated  formulas,  and  by  tak- 
ing pressures  and  temperatures  at  the  summit 
and  some  other  remote  point.  As  an  example, 
a  few  years  ago  the  determination  of  the  height 
of  Alt.  \Yhitney  in  California  was  attempted  by 
taking  pressures  and  temperatures  simultaneous- 
ly at  the  summit  and  at  San  Francisco,  and  then 
applying  Laplace's  formula. 

Some  distinguished  mountain  climbers  have, 
within  the  last  few  years,  made  the  conquest  of  a 
number  of  very  high  summits,  but  unfortunately 
their  height  measurements  may  vary  consider- 
ably from  the  truth,  from  the  fact  that  they  have 
used  incorrect  methods  and  formulas.  It  is  to. 
be  hoped  that  the  present  methods  and  formulas 
will  become  more  generally  known  and  adopted, 
as  all  others  are  a  waste  of  time,  if  accuracy  is 
desired. 

412365 


PREFACE  TO  THE   SECOND   EDITION 

Common  logarithms  are  used  in  the  computa- 
tion, p  must  be  reduced  to  freezing,  and  also, 
when  possible,  to  Paris.  Practically,  the  local 
value  of  g  is  usually  unknown,  but  the  error 
from  neglecting  this  correction  is  slight,  and  it 
may  in  general  be  ignored.  The  correction  for 
capillarity  is  applied  to  all  readings.  The  deci- 
mals may  be  shortened  without  much  error :  thus, 
for  quick  work,  .00003815  gives  nearly  the  same 
result  as  the  longer  decimal. 

Barometers  should,  if  possible,  be  compared 
with  a  standard  barometer,  both  before  and  after 
an  ascent,  f  and  p  must  be  both  taken  in  mms. 
of  mercury,  or  inches  of  mercury. 


PREFACE. 


THE  discrepancies  arising  in  the  calculation  of 
mountain  heights  by  the  barometrical  formulae 
which  have  hitherto  been  in  use  have  brought 
this  valuable  and  in  many  cases  only  applicable 
method  into  disrepute.  The  fault  has  lain  in  the 
formulae,  not  in  the  method,  which  is  one  sus- 
ceptible  of  great  accuracy.  These  formulae  have 
either  been  based  upon  unwarrantable  assump- 
tions or  have  failed  to  take  account  of  all  the  con- 
ditions  obtaining  in  the  problem. 

The  present  essay  \vas  originally  entered  in 
the  Hodgkins  Prize  Competition,  held  under  the 
auspices  of  the  Smithsonian  Institution,  and  was 
awarded  honorable  mention.  In  it  the  important 
problem  of  Barometrical  Hypsometry,  which  has 
not  been  touched  upon  since  1851,  when  it  was 
discjssed  by  Guyot,  has  been  gone  over  anew 
and  brought  up  to  date.  Important  errors  in  the 
older  formulae  have  been  detected  and  a  new 
method  has  been  furnished  which  is  rigidly  accu- 
rate in  theory  and  which  in  practice  will  give  re- 
liable results  under  all  conditions. 

F.  J.  B.  C. 


OTHER  WORKS  BY  F.  J.  B.  CORDEIRO 


THE  ATMOSPHERE,  Its  Characteristics  and  Dynamics, 
viii+129  pages,  35  illustrations,  10^4  x  ll/2  in.,  paper 
binding,  $1.50.  Cloth,  $2.50. 

THE  GYROSCOPE,  Its  Theory  and  Applications,  vii-f-105 
pages,  19  illustrations,  8*4  x6^  in.,  cloth,  $1.50. 

,  THE    MECHANICS    OF    ELECTRICITY,     vi+78    pages,    7 
illustrations,  7^x5J4  "!••  cloth,  $1.25. 


THE 

BAROMETRICAL  DETERMINATION 
OF   HEIGHTS. 


ONE  of  the  most  important  applications  of  the 
known  properties  of  air  has  been  a  deduction 
from  them  of  a  means  of  finding  the  vertical 
height  between  any  two  points,  and  the  problem 
of  measuring  the  vertical  distances  between  any 
two  levels  is  one  that  has  engaged  the  attention 
of  a  number  of  mathematicians  and  physicists  for 
many  years. 

Laplace,  in  the  "  Mecanique  Celeste,"  gave  what 
at  the  time  was  considered  a  complete  solution  of 
the  problem  ;  but  as  it  was  based  upon  several 
unwarrantable  assumptions,  and  took  no  account 
of  the  aqueous  vapor  in  the  atmosphere,  it  was 
at  best  an  approximation. 


2          Barometrical  Determination  of  Heights, 
The   complete    formula    as    given    by    him    is 


Z=  log  £18336. 


(l  -p;  .0028371  cos.  2  L) 

(log  rj+  .868589)  — 

(I    + 

h 


where  Z  =  the  difference  of  level  in  metres  ; 

a  —  Earth's  mean  radius  =  6,366,200  metres; 
L  =  mean  latitude  of  the  two  stations. 

And  further 

(  (  h  =  height  of  barometer  ; 

I   Lower   \  T  =  temperature  of  barometer ; 
(  t    —  temperature  of  air  ; 
At  station   •{ 

II  h'  =  heiglit  of  barometer  ; 
Upper    <  T'  =  temperature  of  barometer  ; 
[_  (  t     —  temperature  of  air  ; 

—  T 

and  I-I  =  h  +  h'" 


The  first  parenthesis  in  the  terminal  factor  is 
the  correction  for  the  difference  of  temperature 
of  the  two  levels.  It  assumes  that  the  problem 
would  be  the  same  if  the  air  between  the  two 
levels  were  of  a  uniform  temperature — the  mean 
of  what  is  observed  at  the  two  levels.  As  a 
matter  of  fact,  if  the  two  stations  are  remote,  a 
large  range  of  temperatures  may  be  found  at  in- 
tervening points. 


Barometrical  Determination  of  Heights.          3 

The  second  parenthesis  is  the  correction  for  the 
change  of  gravity  with  the  latitude.  It  assumes 
that  gravity  increases  regularly  according,  to  a 
law  as  we  go  from  the  equator  to  the  poles  —  a 
supposition  which  we  now  know  to  be  true  only 
in  a  general  way.  The  third  parenthesis  is  the 
correction  for  the  decrease  of  gravity  in  a  vertical 
direction.  It  is  based  upon  the  Newtonian  law 
that  externally  to  the  earth's  surface  gravity  de- 
creases inversely  as  the  square  of  the  distance 
from  the  centre  of  mass.  From  careful  pendu- 
lum experiments  we  know  that  such  a  law  does 
not  hold  near  the  earth's  surface,  large  masses  of 
matter  in  different  localities  causing  variations 
that  are  not  to  be  accounted  for  by  any  simple' 
law. 

Baily,  in  his  "  Astronomical  Tables  and  For- 
mulae," gives  the  following  formula: 

X  -  60345.51  -|  i  +  .ooiiin  (t  +  t'-64°)  j- 

-         X      '  +-002695  cos.  2* 


where  <$>  =  latitude  ; 

/8  =  height  of  barometer  ;      } 

T  =  temperature  of  mercury;  '•  at  lower  station. 

t   —  temperature  of  air;          ) 


#'  =  height  of  barometer  ;      ) 
T'  —  temperature  of  mercury;  I  at 
t'  =  temperature  of  air  ;         ) 


upper  station. 


Feet,  inches,  and  the  Fahrenheit  scale  are  here 
used.     Here  the  same  assumption  is  made  in  re- 


4          Barometrical  Determination  of  Heights. 

gard.to  the  increase  of  gravity  with  the  latitude 
as  in  Laplace's  formula,  and  no  account  is  taken 
of  the  moisture  in  the  air. 

Bessel  first  introduced  in  his  formula,  Astro- 
nomische  Nachrichten,  No.  356,  a  separate  cor- 
rection for  the  effects  of  moisture.  Laplace's 
barometrical  coefficient  is  retained,  but  the  cor- 
rection for  change  of  gravity  is  considerably 
modified. 

Elie  Ritter  in  his  formula,  "Memoires  de  la  So- 
ciete  de  Physique  de  Geneve,"  tome  XIII.,  p.  343, 
gave  a  correction  for  moisture.  The  values  of  the 
barometrical  and  thermometrical  coefficients  are 
derived  from  Regnault's  determinations,  and  a 
new  method  is  proposed  for  applying  the  cor- 
rection due  to  the.  expansion  of  the  air,  which  is 
made  proportional  to  the  square  of  the  differences 
between  the  observed  temperatures  at  each  sta- 
tion. 

Baeyer's  formula,  Poggendorf's  Anna/en  der 
Physik  und  Chemie,  tome  XCVIII.,  p.  371,  does 
not  belong  to  either  of  the  two  classes  just  men- 
tioned ;  for  while  it  keeps  Laplace's  barometrical 
and  thermometrical  coefficients,  it  corrects  the 
effects  of  temperature  by  a  method  analogous  to 
that  of  Ritter,  and  it  entirely  neglects  the  effect 
of  aqueous  vapor. 

Plantamour  in  his  tables  substitutes  for  La- 
place's barometrical  coefficient  that  derived  from 
the  probably  more  accurate  determination  of  the 
relative  weight  of  air  and  mercury  by  Regnault, 


Barometrical  Determination  of  Heights.          5 

viz.,  18404.8  metres.  Laplace  used  the  results  of 
Biot  and  Arago,  and  the  coefficient  deduced  from 
it  was  1 83 1 7  metres.  This  coefficient  was,  however, 
empirically  increased  to  18336  metres  in  order  to 
adjust  the  results  of  the  formula  to  those  fur- 
nished by  the  careful  trigonometrical  measure- 
ments made  by  Ramond  for  the  purpose  of  test- 
ing its  correctness. 

An  error  in  all  these  formulae  was  the  assump- 
tion of  an  absolutely  invariable  barometric  coef- 
ficient. The  barometric  coefficient  in  Baily's 
formula,  60345.51,  varies  under  different  condi- 
tions of  pressure,  temperature,  and  relative  hu- 
midity, from  55,000  to  65,000,  and  in  fact  the  chief 
thing  to  do  before  applying  any  formula  is  to 
compute  exactly  the  barometric  coefficient.  The 
barometric  coefficient,  18336  metres,  which  was 
substituted  in  Laplace's  formula  in  order  to  make 
the  data  of  some  observations  conform  to  Ra- 
mond's  trigonometrical  determinations,  would 
have  had  to  be  changed  if  other  observations  had 
been  taken  under  different  conditions  of  the  at- 
mosphere. 

We  shall  demonstrate  this  in  what  follows, 
elucidating  the  problem  at  each  step,  so  that  it 
may  be  easily  understood  by  any  reader  having 
a  slight  knowledge  of  mathematics. 

If  the  earth  were  a  mere  level  plane,  the  verti- 
cal distance  above  it  of  any  point  would  be  a 
simple  matter  to  determine  ;  but  the  earth  is  not 
a  sphere,  nor  a  spheroid.  Strictly  it  is  only  ap- 


6          Barometrical  Determination  of  Heights. 

proximately  a  regular  figure.  If  the  sea  were  at 
rest,  a  figure. very  nearly  corresponding  to  its  sur- 
face would  be  an  ellipsoid  of  revolution,  having  an 
equatorial  semi-diameter  of  20,926,200  feet,  and  a 
polar  semi-diameter  of  20,854,900  feet,  giving  an 


eliipticity  of  293,465.* 

From  a  comparison  of  the  different  measured 
arcs  of  meridians,  Colonel  Clarke  found  that  the 
surface  most  nearly  agreeing  with  the  sea-level  is 
an  ellipsoid  (not  of  revolution)  having  for  its  equa- 
torial section  an  ellipse  with  a  major  semi-axis  at 
8°  \V.  Ion.  of  20,926,629  feet,  and  a  minor  semi- 
axis  of  20,854,477  feet. 

If  the  air  also  were  at  perfect  rest  it  would  dis- 
tribute itself  about  the  earth  under  the  influence 
of  gravity,  so  that  we  could  trace  out  in  it  sur- 
faces of  equal  density,  or  in  fact  equipotential 
surfaces,  and  these  surfaces  would  be  more  or 
less  similar  and  similarly  situated  to  the  surface 
of  the  earth  as  just  considered.  In  the  strictest 
sense,  then,  the  height  of  a  point  above  the  sur- 
face of  the  earth  would  be  the  perpendicular  dis- 
tance between  the  standard  equipotential  surface 
and  the  point.  Practically,'  however,  in  measur- 
ing heights,  we  determine  the  perpendicular  dis- 
tance between  the  equipotential  surface  passing 
through  the  upper  point  and  that  passing  through 
some  lower  point  of  reference. 

*  Colonel  Clarke,  Geodesy,  p.  319. 


Barometrical  Determination  of  Heights.          7 

In  measuring  heights  we  have  three  methods  at 
our  disposal  : 

I.     That  by  levelling. 
IT.     That  by  vertical  angles. 

III.  That  by  weighing  a  vertical  column  of  air 
between  the  levels,  /.<?.,  the  barometric  method. 

Without  entering  into  a  discussion  of  the  com- 
parative merits  of  the  different  systems,  it  may 
be  stated  that  in  determining  mountain  heights 
with  any  degree  of  accuracy,  the  barometric 
method  is  usually  the  only  practicable  one.  In 
determining  the  height  of  a  mountain  like  Chim- 
borazo,  as  Mr.  Whymper  did,  it  is  wholly  out  of 
the  question  to  level  it,  and  vertical  angles  taken 
at  great  distances  can  only  lead  to  an  approxima- 
tion. 

The  determination  of  a  height  by  a  barometer 
is  equivalent  to  the  weighing  of  a  column  of  air 
under  varying  conditions.  Reduced  to  its  sim- 
plest terms  the  atmosphere  is  supposed  to  be  ab- 
solutely at  rest,  of  a  uniform  temperature  tlnough- 
out,  absolutely  dry,  and  the  value  of  g — the  force 
of  gravity  at  the  place — to  be  known.  Then  it 
will  be  easy  to  obtain  the  weight  of  a  vertical  col- 
umn of  air  between  any  two  points,  and  thereby 
the  height  of  this  column. 

By  Mariotte's  law  we  know  that  the  density  of 
air  is  proportional  to  the  pressure  to  which  it  is 
subjected.  If  we  know  accurately  the  weight  of 
each  cubic  foot  of  air  in  the  column,  the  pressure 
of  the  whole  column  per  square  foot  will  be  ar- 


8          Barometrical  Determination  of  Heights. 

rived  at  by  simply  adding  the  individual  weights. 
Such  a  summation,  when  the  elements  vary  ac- 
cording to  a  law,  can  be  easily  effected  by  the  in- 
tegral calculus. 

Let  the  weight  of  a  cubic  foot  of  air  which  is 
the  increment  or  increase  of  pressure  for  every 
foot  that  we  descend  be  represented  by  Sp.  Now 
this  weight  or  increment  is  itself  proportional  to 
the  pressure  by  Mariotte's  law  and  also  propor- 
tional to  the  height  of  the  element, -which  is  here 
supposed  to  be  the  unit  of  measure — one  foot. 
Represent  this  element  of  height  by  Sh,  then  Sp 
=  c.p.  Sh,  where  c  is  some  constant. 

Or  Sh  =  k  -—  •  Summing  our  elements,  or  in 
other  words,  integrating,  we  have 


.-.  h  =  k  log  F°  where  p0  is  the  pressure   given 

by  the   barometer  at  the    lower   station,    px  the 
pressure  at  the  upper  station  and  h  is  the  height. 

If  we  let  h  =  i,  then  k=  — 


where  w  represents  the  weight  of  a  cubic  foot  of 
air,  expressed,  of  course,  in   the   same   terms  as 


Barometrical  Determination  of  Heights.          9 

the  pressure.  If  then  we  determine  once  for  all 
the  weight  of  a  cubic  foot  of  air  at  the  pressure 
p,  for  the  temperature  t  and  value  g  of  gravity, 
we  have  a  value  of  k  which  can  be  used  in  all 
calculations  where  the  conditions  remain  the 

same.       But    the    above    formula,  h  =  k  log  - 

holds  only  provided  the  conditions  remain  the 
same  throughout  the  whole  column  of  air — a 
state  of  affairs  that  never  occurs  in  nature  except 
for  narrow  limits.  In  ascending  from  one  level  to 
another  we  pass  through  many  varying  condi- 
tions. One  layer  of  the  atmosphere  may  be  much 
warmer  than  another  immediately  above  or  be- 
low it. 

The  relative  humidity  varies  from  point  to 
point  and  from  hour  to  hour.  The  value  of  g  does 
not  vary  greatly  for  different  levels  at  the  same 
place,  but  varies  sufficiently  at  places  widely 
removed  from  each  other  on  the  earth's  surface 
to  be  taken  into  account.  Violent  commotions 
of  the  atmosphere  are  another  disturbing  factor 
on  the  pressure,  the  swirls  and  eddies  of  a  storm 
often  causing  the  barometer  to  jump  up  and 
down  a  hundredth  of  an  inch  or  more.  Under 
these  conditions  the  attempt  to  weigh  a  long 
column  of  air — to  determine  a  great  height — by 
taking  conditions  (pressure,  temperature,  and 
relative  humidity)  at  the  base  and  summit  of  a 
mountain  simultaneously,  and  then  taking  the 
mean  of  the  latter  two  is  futile,  although  baromet- 


io       Barometrical  Determination  of  Heights. 

rical  formulae  have  usually  been  applied  in  this 
manner. 

There  remains,  however,  a  means  by  which  a 
very  close  estimate  can  be  arrived  at  in  spite  of 
all  these  obstacles.  That  is  to  take  the  condi- 
tions (t,  p,  and  r.h.)  at  frequent  intervals,  so  that 
we  can  be  reasonably  sure  that  from  one  point  to 
another  the  conditions  of  our  original  formula 
hold  to  a  close  approximation. 

The  occasions  are  very  rare,  especially  when 
the  weather  is  at  all  settled,  where  any  very  sud- 
den changes  take  place  for  an  ascent  of  500  feet 
or  more.  But  even  then  it  is  easy  to  multiply 
stations  and  take  the  readings  of  the  barometer, 
thermometer,  and  psychrometer.  Our  total 
height  will  then  be  the  algebraic  sum  of  our  ele- 
ments. As  a  correction  we  can  take  readings 
again  on  descending  and  average  the  heights  by 
ascent  and  descent. 

All  this  necessitates  the  separate  determination 
of  the  constant  k  for  each  interval  of  ascent,  and 
this  requires  that  besides  knowing  the  extreme 
pressures  we  also  compute  the  weight  of  a  cubic 
foot  of  air  as  found  either  at  the  top  or  bottom  of 
the  interval. 

The  mercurial  barometer  is  the  only  instrument 
at  £he  present  time  that  we  can  rely  upon  for 
taking  pressures.  The  aneroid  barometer  is  a 
misnomer  ;  it  should  be  called  the  aneroid  baro- 
scope. The  best  of  these  instruments  are  entire- 
ly unreliable  and  irregular  in  their  action.  Mr. 


Barometrical  Determination  of  Heights.        n 

Whymper  ("  How  to  Use  the  Aneroid  Barom- 
eter ")  has  done  an  important  service  in  showing 
conclusively  that  they  are  worse  than  useless,  as 
they  always  underread  the  mercurial  barometer, 
and  in  a  most  irregular  way  at  that. 

The  mercurial  barometer  should  be  suspended 
from  a  tripod  and  exposed  to  the  atmosphere  for 
about  ten  minutes,  when  several  readings  should 
be  taken,  and  the  average  entered  as  the  reading 
for  the  station  at  that  time.  The  readings  of  the 
thermometer  and  psychrometer  are  taken  simul- 
taneously. The  height  of  the  mercurial  column 
is  carefully  reduced  afterward  to  freezing,  and 
the  correction  for  capillarity  is  applied.  Where 
great  accuracy  is  aimed  at,  or  where  there  is  rea- 
son to  believe  that  the  value  of  g  at  the  place 
differs  considerably  from  tlie  standard  value — 
that  of  Paris,  for  our  purposes,  since  most  of  our 
data  were  established  at  that  place  by  Regnault — 
we  must  apply  this  correction  also. 

Besides  a  barometer  two  other  instruments  only 
are  required — a  thermometer  and  a  hygrometer. 
The  thermometer  attached  to  the  barometer  is 
sufficient  to  determine  the  temperature  of  the  at- 
mosphere. 

Regnault  found  that  the  weight  of  a  litre  of  dry 
air  at  Paris  under  a  pressure  of  760  mm,  of  mer- 
cury was  1.293233  grammes.  This  figure  is  the 
base  upon  which  all  our  calculations  rest.  The 
force  of  gravity  at  Paris  being  gp — 32.1747  foot- 
seconds,  or  980.94  dynes — the  standard  of  height 


12        Barometrical  Determination  of  Heights. 

of  our  barometer,  760  mm.,  will  vary  with  the 
force  of  gravity.  Consequently,  the  weight  of  a 
litre  of  dry  air  at  any  other  place  x  will  be 

cr 

I-293233  -—  grammes. 


The  value  of  g  has  been  determined  with  a  con- 
siderable degree  of  accuracy  at  various  points  of 
the  earth's  surface.  Where  the  value  has  been 
determined  near  some  place  where  the  altitude 
is  to  be  determined,  it  is  best  to  rely  upon  that 
value.  Laplace's  and  the  other  formulae  assume 
that  this  value  varies  according  to  a  regular  law 
with  the  latitude,  and  also  with  the  altitude  above 
the  sea-level.  Actually  such  a  law  exists  only  to 
a  very  limited  extent. 

While  it  is  probable  that  the  decrease  of  gravity 
as  we  ascend  directly  into  the  air  (/.<?.,  by  bal- 
loon), would  conform  more  or  less  nearly  to  the 
law  of  the  inverse  squares,  such  a  law  is  by  no 
means  the  case  on  a  mountain-slope.  Menden- 
hall  found  that  the  value  of  g  on  the  summit  of 
Fuji-yama  was  greater  than  at  the  base.  On 
many  mountains  it  may  be  slightly  less  at  the 
top,  but  wanting  definite  measurements  it  would 
be  safer  to  assume  that  there  is  no  appreciable 
change. 

The  following  table  will  give  some  idea  of  the 
variability  of  g. 


Barometrical  Determination  of  Heights.        13 


TABLE  I. 

VARIABILITY   OF  GRAVITY  AT: 


Greenwich,  g  =  32.1912  )  ,    .  A 

Paris,  1  =  32.1747  J-  feet  -seconds. 

Washington,  g  =  980. 100  ~] 

Mt.  Hamilton,  Cal.,  g  =  979.651   I 

Honolulu,  H.  I.,       g  =  978.936  I 

Waikiki,  H.  I. ,          g  =  978.922  }•  dynes. 

Kawaihfe,  H.  I.,       g  =  978.803   I 

Kalaieha,  H.  I.,        g  =  978490  |    —6660 ft. 

Mauna  Kea,  H.  I.,  g  =  978.060  J    —summit. 

The  last  three  determinations  are  for  the  slopes 
of  Mauna  Kea,  the  first,  Kawaihae,  at  the  base,  the 
second,  Kalaieha,  half  way  up,  and  the  third  at 
the  summit.  In  general,  the  value  of  g  is  less  on 
islands  than  on  continents. 

Owing  to  the  ellipticity  of  the  earth  and  the 
effect  of  centrifugal  force  becoming  less  as  we  ap- 
proach the  poles,  gravity  tends  to  increase  from 
the  equator  toward  the  poles.  Let  the  elliptic 
section  through  the  poles  be  represented  by 

r'2cos.2a     r2  sin.  2a  a2  b2 

a2 b2 ~  r  '  '•  r  =  b2  +  c 2  sin.2  a 

By  the  law  of  inverse  squares, 
k  k 

g  =-2  . '.  g  =  ^j  (b2  +  c  2  sin.2  a)  =  k'  +  k"  sin.2  a 

Therefore,  the  increment  of  g  from  the  equator 
to  the  poles  is  proportional  to  the  square  of  the 
sine  of  the  latitude.  Considering  the  effect  of 
centrifugal  force  we  have  C.  F  —  K  cos.  a.  Re- 
solving this  in  the  direction  of  gravity  its  result- 


14        Barometrical  Determination  of  Heights. 

ant  becomes  K  cos.2a  .•.  g  (the  resultant  force  of 
gravity)  =  f  (actual  force)  —  K  (i  —  sin.2  a)  .*.  g 
=  K'  +  K"  sin.'V 

Thus  the  increase  of  g  from  diminution  of  the 
centrifugal  force  varies  according  to  the  same  law 
and  we  can  combine  the  formulae  thus:  g  =  ge 
(i  +.005133  sin.2  L),  where  ge  is  the  force  of 
gravity  at  the  equator.  An  approximate  value 
of  ge  is  32.088,  but,  as  we  have  seen,  it  is  not  con- 
stant. The  formula  given  above,  with  the  coeffi- 
cient .005133,  was  deduced  originally  by  Clairaut. 
Measurements  show  that  gravity  conforms  to  a 
certain  degree  with  this  law,  and  it  is  advisable 
where  definite  determinations  are  wanting  to  ap- 
ply such  a  correction.  It  must  be  borne  in  mind, 
however,  that  gravity  may  be  less  on  an  island 
situated  far  to  the  north  than  at  a  continental 
point  near  the  equator. 

The  weight  of  a  litre  of  pure  mercury  Regnault 
found  to  be  13596  grammes.  Let  us  suppose  that 
at  Paris,  the  air  being  perfectly -dry,  the  tempera- 
ture is  o°  Cent,  and  the  pressure  p  expressed  in 
inches  of  mercury.  By  our  barometrical  formula 

i 
k  =  p  +  w,  where  w  is  the  weight  of  a  cubic 

foot  of  air  expressed  in  inches  of  mercury.  There- 
fore, 


Barometrical  Determination  of  Heights,        15 

But  suppose  the  temperature  and  relative  hu- 
midity are  not  zero.  As  the  temperature  increases 
the  weight  of  a  cubic  foot  of  air  becomes  less  for 
ihe  same  pressure.  The  coefficient  of  expansion 
of  air  and  most  gases  is  .00367  per  degree  Cent. 


vvt= 


I  +  a  t    '  I  +  .00367 


The  relative  humidity  also  affects  the  weight  of  a 
given  volume  of  air.  Moist  air  is  always  lighter 
than  dry  air.  The  relative  humidity  is  defined 
to  be  the  ratio  of  the  pressure  of  the  vapor  in  a 
given  volume  of  air  to  the  pressure  of  the  vapor 
in  the  same  volume  at  saturation. 

Let  V  denote  a  volume  of  air,  H  its  pressure, 
f  the  pressure  of  the  vapor  in  it  and  t  the  tem- 
perature. Tiie  entire  gaseous  mass  may  be  di- 
vided into  two  parts,  a  volume  V  of  dry  air  at 
temperature  t  and  pressure  H— f,  the  weight  of 
which  is 

i  H-f 

V   x   1.20^2-?^   x   -  -  x—          — »    and    a    vol- 

i  -f    a  t  760 

ume  V  of  aqueous  vapor  at  the  temperature  t  and 
pressure  f.  Since  the  density  of  aqueous  vapor  is 
f  that  of  dry*  air,  the  weight  of  this  latter  mass  is 

J  V  x  1.293233  x  -X 


i  Hh  a  t       760 
The  sum  of  these  two  weights  is  the  weight  re- 
quired, viz.  :  V  x  1.293233  X 


i  -t-  a  t 


16     Barometrical  Determination  of  Heights. 

The  pressures  in  mm.  of  mercury  at  satura- 
tion for  different  temperatures  of  aqueous  vapor 
have  been  carefully  determined  by  Regnault  and 
are  given  in  the  following  table  : 

TABLE  II. 

PRESSURES    IN    MM.  OF    MERCURY    AT    SATURATION  FOR  DIF- 
FERENT   TEMPERATURES    OF    AQUEOUS    VAPOR. 


Temperatures, 
Cent. 

Force  of 
vapor,  mm. 

Temperatures, 
Cent. 

Force  of 
vapor,  mm. 

—  32 

0.32 

10 

9.17 

—  2O 

•93 

'5 

12.70 

—  10 

2.08 

20 

17-39 

—   5 

3*" 

.        25 

23-55 

0 

4.60 

30 

31-55 

5 

6.53 

40 

5Ll6 

A  more  complete  table  is  given  at  the  end. 

The  force  of  the  vapor  in  the  atmosphere  is 
obtained  by  observing  the  dew-point,  which 
gives  us  the  temperature  at  which  saturation  oc- 
curs for  the  given  pressure.  The  actual  force  of 
the  vapor  will  be  the  tension  of  saturated  vapor  at 
the  dew-point.  An  efficient  instrument  for  deter- 
mining the  dew-point  is  Dines's  Hygrometer.  Or 
the  relative  humidity  may  be  obtained  from  ta- 
bles by  readings  of  the  wet  and  dry  bulb  hygrom- 
eter. The  results  of  this  latter  instrument, 
sometimes  known  0s  August's  Psychrometer,  are 
far  from  accurate. 

We  can  thus,  from  the  foregoing  formula,  de- 
termine the  weight  of  a  cubic  foot  of  moist  air 


Barometrical  Determination  of  Heights.        17 

under  any  given  conditions  of  temperature,  pres- 
sure, and  humidity. 

At  the  pressure  p,  reduced  to  Paris,  tempera- 
ture t  and  vapor  tension  f,  the  weight  of  a  cubic 
foot  of  moist  air,  expressed  in  inches  of  mercury, 
is 

12  I  P— If 

•^^.^^..-.vx  vx  \X"o 


i.zy^zjj  A  - 
P-ff 

13596     '    i  +.003671 
x  .0000381468. 

i 

29.9218 

i  +  ,oo367t 
k  — 

log(P 

p  —  f  f 
X   oooo 

381468^ 

i  4-  .003671 

where  t  is  Centigrade  and  p  expressed  in  inches 
of  mercury  reduced  to  Paris  and  freezing,  and 
corrected  for  capillarity.  The  corrections  for  ca- 
pillarity and  freezing  are  taken  from  carefully 
prepared  tables  which  are  easily  obtained.  The 
key,  then,  to  all  our  barometric  determinations 
is  the  computation  of  the  coefficient  k.  Hav- 
ing obtained  this  for  a  range  where  the  varia- 
tions of  this  factor  are  inappreciable,  it  remains 
only  to  get  the  height  by  applying  the  formula 

h  =  k  log  -2s-  • 

The  computations  are  made  after  the  ascent 
from  the  data  collected.  These  can  be  taken  with- 
out much  trouble  during  the  ascent  and  descent. 
The  readings  should  be  carefully  recorded  in  a 


1 8        Barometrical  Determination  of  Heights. 

systematic  manner,  so  that  no  doubt  can  apper- 
tain to  them.  Asa  rule,  one  man  should  carry  the 
barometer  or  barometers,  and  nothing  else.  He 
should  be  intrusted  with  the  readings  of  the  mer- 
curial column  and  the  thermometer  attached. 
Another  man  should  carry  and  record,  simulta- 
neously, the  hygrometer.  At  the  point  of  depart- 
ure— the  station  from  which  the  heights  are  to  be 
estimated — the  tripod  should  be  set  up,  the  ba- 
rometer suspended,  and  after  exposing  it  to  the 
air  for  ten  minutes  or  more,  to  make  sure  that  the 
thermometer  attached  and  all  the  parts  of  the  ba- 
rometer are  at  the  same  temperature  as  the  air, 
the  readings  should  be  taken.  At  the  same  time 
the  hygrometer  should  be  read.  The  body  of  the 
observer  should  not  be  too  close  to  the  instru- 
ments. It  is  well  to  take  several  readings  at  each 
station  and  average  them.  Whenever  in  the  as- 
cent (or  descent)  it  becomes  evident  that  the 
conditions  are  changing,  a  stop  should  be  made, 
station  noted,  and  readings  taken.  Whether  the 
conditions  are  changing  or  not  can  be  determined 
from  time  to  time  without  a  stop  by  simply  noting 
the  temperature  or  glancing  at  the  wet  and 
dry  bulbs,  if  a  psychrometer  is  taken.  When- 
ever a  prolonged  stop  is  made,  as  at  dinner  or  on 
camping  for  the  night,  the  readings  should  be 
taken  directly  after  the  halt  and  again  on  starting 
out,  taking  this  station  as  a  new  point  of  depart- 
ure. It  is  possible  that  a  long  ascent  may  be 
made  without  a  perceptible  change  of  conditions, 


Barometrical  Determination  of  Heights.        19 

&nd  in  this  way  tedious  computations  may  be 
avoided. 

Such  is  the  problem  of  measuring  heights  by  the 
barometer — a  process  perfectly  intelligible  at  every 
step,  and  consisting  of  weighing  every  layer  of  the 
atmosphere  through  which  the  ascent  is  made. 

Since  in  a  mixture  of  one  or  more  gases  in  a 
given  space,  each  gas  behaves  dynamically  the 
same  as  if  it  occupied  the  space  alone,  it  is  evi- 
dent that  where  the  conditions  remain  the  same, 
we  can  compute  heights  by  three  different  sets  of 
data.  We  can  consider  the  air  as  a  mixture  of 
dry  air  and  vapor,  as  has  been  done  in  the  fore- 
going discussion,  using  the  formula  h  =  k  log  — 

Px 

where  k  is  derived  from  the  weight  of  a  cubic  foot 
of  moist  air  under  the  obtaining  conditions,  and 
p0  and  px  are  the  actual  barometric  pressures  ob- 
served. Or  we  may  weigh  a  column  of  dry  air  be- 
tween the  two  stations,  using  the  formula  h  =  k' 

log  J    ___2   where    p0  and  px  are   the  barometric 

Px        *x 

readings,  and  f0  fx  are  the  vapor  tensions  at  the 
extreme  stations,  while  k'  is  derived  from  the 
weight  of  a  cubic  foot  of  dry  air  at  pressure 
p0 —  f0  and  temperature  t. 

Again,  making  use  of  the  vapor  alone,  we  can 

employ   the    formula  h  =  k"  log  -•-  ,  where  k"  is 

*x 

derived  from  the  weight  of  a  cubic  foot  of  vapor  at 
the  pressure  f0  and  temperature  t.  The  last  two 


2o        Barometrical  Determination  of  Heights. 

methods  will  not  be  so  reliable  as  the  chief 
method,  since  the  tension  f,  which  is  derived  from 
the  dew-point,  cannot  be  measured  so  closely  as 
the  pressure  p  by  a  good  barometer.  Still,  all 
these  methods  may  be  employed  as  checks  on  the 
others. 

A  practical  example  is  appended  illustrating 
the  method  in  practice  and  showing  within  what 
limits  of  accuracy  heights  can  be  thus  determined. 
A  seven-place  table  of  logarithms  will  be  advis- 
able for  the  computations. 

Barometer  at  lower  station  read  30.2784,  and  at 
upper  station  30.224,  corrected  for  gravity,  but 
not  for  capillarity  and  temperature.  Tempera- 
ture at  both  stations  15°  C.,  and  f  at  lower  station 
was  .2434  inch.  The  error  of  the  barometer  was 
very  slight,  if  any. 

30.2784 

—  .085    correction  for  freezing  point. 

30.1934 
+     .037    correction  for  capillarity. 

30.2304  corrected  reading  for  lower  station. 

30.224 

—  .085    correction  for  freezing  point. 

correction  for  capillarity. 


30.176    corrected  reading  for  upper  station. 

f  f  =  .091  inch. 

.'.  p  —  |  f  =  30  176  —  .091  —  30.085 

I  -I-  .00367  t=  1.055 

log  30.085  =  1.4783500 

log  1.055  =  °-0232525 

log  .0000381468  =  5.5814527 

.'.log   TV  "03671  x  -000038 1 468  =  ^0365502 


Barometrical  Determination  of  Heights.         21 

This  logarithm  corresponds  to  .001087  inch. 
This  shows  that  under  the  conditions  a  cubic  foot 
of  air  weighed  .001087  inch  of  mercury. 

P  +  .001087  =  30.1771  — 
log  30.  1771=   i.4796775 

log  30.  176  —  1.4796617 
.0000158 


=  63291.1  =  k 

log  px  =  1.4804438 
log  p2  =  1.4796617 

.0007821  x  k  = 

It  will  be  noticed  that  the  last  four  logarithms, 
though  actually  natural  logarithms  in  the  for- 
mulae, are  here  computed  as  common  logarithms, 
since  the  modulus  cancels  out. 

Therefore  the  height  between  the  two  stations 
is  by  the  computations  49.5  feet.  The  height  as 
previously  determined  by  levelling  was  50  feet. 
A  number  of  determinations  of  the  same  height 
were  made  under  widely  differing  conditions  of 
the  atmosphere  and  the  results  all  came  out  within 
half  a  foot  of  the  correct  height,  50  feet  —  some 
of  them  being  remarkably  close. 

Such  short  distances  are  very  severe  tests  for  a 
barometer.  Where  the  heights  are  greater  the 
proportional  error  will  be  less.  By  no  refinement 
is  it  possible  to  measure  with  a  barometer  much 
less  than  a  foot,  since,  as  we  have  seen,  the  weight 
of  a  cubic  foot  of  air  is  equivalent  to  only  .001 
(4-  or—  )  inch  of  mercury,  and  our  barometers 
do  not  read  finer  than  thousandths.  By  careful 


22        Barometrical  Determination  of  Heights. 

work  it  is  probable  that  our  error  need  never  ex- 
ceed 5  feet  in  1000.  By  our  method  also  of  mul- 
tiple stations,  the  errors,  which  are  as  liable  to 
be  above  as  below,  will  be  apt  to  eliminate 
themselves  in  the  summation. 

The  altitude  of  Mont  Blanc  was  computed  by 
Delcros,  from  measurements  taken  by  MM.  Bra- 
vais  and  Martins,  to  be  4810  metres,  which  came 
strikingly  close  to  the  result  obtained  by  a  geo- 
detic survey,  viz.  :  4809.6.  But  such  a  coinci- 
dence must  not  deceive  us.  It  is  a  mere  coinci- 
dence, as  Delcros's  formula  is  not  susceptible  of 
such  accuracy — particularly  as  it  wholly  neglects 
the  moisture  of  the  air. 


TABLE  III. 

ELASTIC  FORCE  OF  SATURATED  AQUEOUS  VAPOR.  EXPRESSED 
IN  INCHES  OF  MERCURY  AND  FAHRENHEIT  TEMPERATURES. 


Temp. 

Force.  '  Temp. 

Force.  Temp.        Force. 

i 

'-  51 

.0087   13 

•0783  1  57         -4655 

~  30 

.OOQ2 

14 

.0818   58           .4825 

-   29 

.0098 

15 

.0857  59         .5000 

--  28 

.OIO4 

16 

.0898 

60         .5179 

27 

.OIIO 

17 

.0940 

61         .5365 

--  26 

.0117 

18 

.0984 

62        -.5558 

•-  25  - 

.0124  '  19 

.1030 

63         .5756 

-  24 

.0131   20 

.1078  64         -5962 

*  23 

.0138 

21 

.1128  65         .6173 

•—  22 

.0146 

22 

.1179  66         .6392 

—  21 

.0154 

23 

.1233  67         .6616 

»«  20 

.0163 

24 

.1289  68         .6847 

—  19 

.OI7I 

25 

.1347  69         .7084 

-  18 

.Ol8l 

26 

•1407  70         .7329 

-  -7 

.OigO 

27 

.1469  71         .7583 

—  16 

.O2OO 

28 

.1534  72         .7844 

—  15 

.0210 

29  ' 

.1600  73         .8113 

-  14 

.O22I 

30 

.1668   74         -8391 

-  13 

.O232 

31 

•1739  75         -8676 

—  12 

.0244 

32 

.1811   76         .8970 

—  II 

.0257 

33 

.1883  77         .9272 

—  IO 

.0270 

34 

.1959  78         .9582 

-  9 

.0283 

35 

.2037  79         .9902 

g 

.0297   36 

.2119  So 

.0232 

7 

.0312  i  37 

.2204  Si 

.0572 

-  6 

.0327  38 

.2291  82 

.0922 

-  5 

'•0343  |  39 

.2382  83 

.1281 

-  4 

•0359  40 

.2476 

84 

.1651 

-  3 

.0376  41 

•2572  85 

.2031 

-*   2 

•°395 

42 

.2672  86 

.2421 

—   I 

•0454  43 

.2775  87 

.2821 

O 

.0434  44 

.2882  :  88 

•  3234 

I 

•0454  45 

.2993  89 

•  3659 

2 

.0476 

46 

.3108 

90 

.4097 

3 

.0498 

47 

-3228 

91 

•  4546 

4 

.0521 

48 

3.35* 

92 

.5008 

5 

•0545 

49 

•3477 

93 

.5482 

6 

.0570 

50 

.3608 

94 

.5969 

7 

•0597 

51 

•3743 

95 

.6468- 

S 

.0625 

52 

.3882 

96 

.6980 

9 

•0654  53 

.4027 

97 

.7508 

IO 

.0684  54 

.4176 

98 

.8050 

ii 

•0716  55 

•4331 

99 

.8607 

12 

.0749  56 

.4490 

100 

.9179 

24    Barometrical  Determination  of  Heights. 


TABLE  IV. 


Pressures  of  Aqueous  Vapor  in   Mms.   of   Mercury 
at  Saturation,  for  Different  Temperatures  Cent. 


Temp. 

Press. 

Temp. 

Press. 

Temp. 

Press. 

Temp. 

Press. 

—10° 

2.0/8 

12° 

10.457 

29° 

29.782     90° 

52545 

8 

2.456 

13 

11.062 

30 

31.548     91 

54578 

6 

2.890 

14 

11.906 

31 

33405 

92 

566.76 

4 

3.387 

15 

12.699 

32 

35-359 

93 

588.41 

2 

3-955 

16 

13.635 

33 

37-410 

94 

610.74 

O 

4.600 

17 

14.421 

34 

39.565 

95 

63378 

4-  i 

4.940 

18 

15-357 

35 

41.827 

96 

657.54 

2 

5-302 

19 

16.346 

40 

51.160 

97 

682.03 

3 

5-687 

20 

I7.39I 

45 

7L39I 

98 

707.26 

4 

6.097 

21 

18.495 

50 

91.982 

98.5 

720.15 

5 

6.534 

22 

19.659 

55 

II7479 

99 

733-91 

6 

6.998 

23 

20.888 

60 

149.321 

99-5 

746.50 

7 

7-492 

24 

22.184 

65 

186.945 

IOO 

760.00 

8 

8.017 

25 

23-550 

70 

233-093:100.5 

773-71 

9 

8-574 

26 

24.998 

75 

288.517 

101 

787.63 

10 

9.165 

27 

26.505 

80 

355-40 

102 

8l6.I7 

IT 

9.792  28 

28.101  85 

433-41 

104 

875-69 

Barometrical  Determination  of  Heights.     25 

STANDARD  BAROMETRICAL  FORMULAS 

h  =  height 


I    _  k  iog. 


PQ  k  =  barometrical   coefficient 

p^  p0  =  pressure  at  lower  station 

p4  =  pressure  at  upper  station 


k  = 


3* 
i  — 


log  (i  +  .000125156. 


i  -f-  .003671 
t  is  Cent.    Gives  height  in  metres. 


k  = 


3f 
i  — 


8P 
log  (i  +  .0000381468. 


) 
_J 


.00204  (t  —  32) 
t  is  Fahr.     Gives  height  in  feet. 


k  = 


i 

8p 
log  (i  -f  .0000381468 . 


i  +  .003671 
t  is  Cent.  Gives  height  in  feet. 


26     Barometrical  Determination  of  Heights. 


NOTE  ON  BOILING  THERMOMETERS. 

By  determining  the  temperature  at  which 
pure  water  boils,  we  can,  from  the  table  giving 
the  tension  of  saturated  vapor  at  various  tem- 
peratures find  the  pressures  of  the  atmosphere. 
If  we  are  provided  with  thermometers  which  can 
be  depended  upon  to  give  the  boiling  point  within 
i/ioo  of  a  degree  Cent,  (which  is  doubtful), 
we  shall  be  able  by  this  method  to  determine  dif- 
ferences of  level  of  about  10  feet,  since  at  lower 
levels,  an  ascent  of  about  1080  feet  makes  a 
difference. of  i°  Cent,  in  the  boiling  point.  But 
by  an  ordinary  mercurial  barometer,  we  can, 
with  less  trouble,  determine  differences  of  level 
of  one  foot.  Hence,  the  mercurial  barometer 
remains  the  most  reliable  and  accurate  instru- 
ment we  possess  at  present  for  the  measurement 
of  pressures. 


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